9 ms·

Vi Hart: Every school should replace calculus with recreational math

daoudc 16y ago
It's so true that the way that maths is taught at schools needs a complete revamp. At secondary school level it should be taught as a creative subject from the start: choose some rules and see what happens.
xyzzyz 16y ago
I'm a Math and CS student and still wonder how was it even possible that I did not start to hate math in high school. Things done there are dull and boring, exercises are repetitive and do not teach anything substantial other than applying another algorithm shown by a teacher. I found out that almost everyone did not really understand not only the problems that were told to solve, but also the solutions they gave -- and it was hard to blame them, for it was mainly teacher's and education system's fault not to give them thorough understanding of what is math really about. My idea? Dump all that algorithmic crap and teach real problem solving. Do not punish for bad performance, because it is really hard to actually come up with a solution to a problem, and this skill is not something that can be learned by doing the same thing over and over again. Problem solving also require less concepts to be introduced and internalized, so more time can be spent on actual thinking and discussing, and less on preparations. This kind of math course may seem not to prepare to good to "university level" math, but in fact it is quite opposite -- universities have to start almost from scratch anyway, and it is easier to teach new concepts a person who has a good understanding what math really is, than a person who only know tens of algorithms, used in high school to grind all kind of "problems" given there.
- btam 16y ago
  The thing is, students already think of math as difficult. If we switched from algorithms to actual creative problem solving, it would become a lot more intimidating. I'm pretty confident of this. I'm in high school, and the other students have noted how much more they would 'hate' math if it were like the creative problem solving math competition that we have here in the US each year. http://en.wikipedia.org/wiki/American_Mathematics_Competitions http://en.wikipedia.org/wiki/American_Mathematics_Competitio... What many -- perhaps even most -- kids really want is a course where they do only simple mechanical work, and get a boost to their GPA. At least, that's what it seems to me to be. Also, we have to consider that unfortunately, not all teachers care. Now that teaching to the test is rampant(at least in the US), I think many classes would fall apart if the added structure and 'accountability' disappeared, like it would if we made that switch. I don't disagree with you. The world would be a better place if we could get rid of this; if we didn't have all of this baggage to deal with. EDIT: RiderOfGiraffes seems to have said the same thing in a different way in this thread; check it out: http://news.ycombinator.com/item?id=2390960 http://news.ycombinator.com/item?id=2390960.
  - RiderOfGiraffes 16y ago
    But that's math competition stuff, which is seriously more daunting, and practising mathematicians will often claim is even less like "real, proper math." It's occasionally suggested that some real, long-standing unsolved problems should be set as IMO questions, because there's a non-zero chance they'd get solved. > What many -- perhaps even most -- kids really want is a > course where they do only simple mechanical work, and > get a boost to their GPA. "Most people would sooner die than think. And most of them do." -- G.B.Shaw. > ... not all teachers care. And not all teachers are capable. And largely all of your points are exactly right, except for your misconception about what kind of math we want kids to be doing. Going back to that point, the IMO is like playing at Carnegie Hall. What we want is like messing about with instruments to see what they do, and getting hooked on the curious things that are possible. I'm not offering solutions here, I'm just helping define the problem, and explore directions.
    - btam 16y ago
      You're pretty much right on the math competitions front, but the first few problems on the AMC are pretty simple IMO(pardon the pun). They are barely above the level of mechanical problem solving. It does get harder, though, and I agree that that is more daunting. You've reminded me, however -- I remembered thinking that the Phillips Exeter Academy model for math education (the "Harkness" method) is amazing. http://en.wikipedia.org/wiki/Phillips_Exeter_Academy#Harkness_teaching_method http://en.wikipedia.org/wiki/Phillips_Exeter_Academy#Harknes... > Exeter does not teach math with traditional textbooks. Instead, > math teachers assign problems from workbooks that have been > written collectively by the Academy's math department. From these > custom workbooks, students are assigned word problems as > homework. In class, students then present their solutions at the > blackboard. This means that in math class at Exeter, students are not > given theorems, model problems, or principles beforehand. Instead, > theorems and principles emerge more organically, as students work > through the word problems."* This is what we want. Discovering math makes it fun for everyone. Honesty and actual learning with that method(my whole would be not be too difficult, because everyone would inevitably have their own personal spin on the origins of the theorems et cetera. Plus, it'd be embarrassing to go to the blackboard and say that you haven't done the work. ---- By the way, expect an email regarding what you said about the IMO.
- crasshopper 16y ago
  Lockhart's Lament: www.maa.org/devlin/LockhartsLament.pdf A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory.... Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music ... are considered very advanced topics and are generally put off until college, and more often graduate school.
- samlittlewood 16y ago
  The incentives should be around changes in performance, +ve or -ve, especially if it was as a result of the kid's own actions.
- icegreentea 16y ago
  From talking to my professors, there really was once a time when university math (calc 101 or w/e) was not starting from scratch (from the perspective of engineering anyways). Spend maybe a month over derivatives and limits, and then off to integrals and everything else you'll need so that mechanics actually makes some sense.
  - deleted 16y ago
    [deleted]
- neutronicus 16y ago
  I'm probably one of the few, but I didn't have any problem with the paradigm of high school math education, just the pace. As far as my later career, it's been my experience that rote manipulation without conceptual understanding can solve your problem, and that the converse isn't true. Sometimes, if you do the math right, it will contradict your intuition. I'm hesitant to endorse any paradigm that aims to make math more intuitive, since at the end of the day that's a lie. One of the take-home messages of mathematics education should be that these mechanical processes are better than your own "problem-solving skills".
  - eru 16y ago
    In some sense, yes. Mathematics is the science of how to solve problems without thinking. Progress in mathematics means elimination of thought needed. But actually inventing that time saving machinery is different from applying it.
    - ajscherer 16y ago
      Mathematics is most definitley not that. The fact that your definition probably sounds reasonable to most people who made it all the way through calculus (and even excelled in those courses) probably supports what Vi is saying. On the other hand, maybe it's okay that people don't really understand what math is (as mathematicians understand it). I don't see any evidence that society is falling apart due to lack of appreciation for pure math.
      - neutronicus 16y ago
        I think "mathematics is a mechanical process for solving problems without an intuitive grasp of their nature" is a very good definition, with which many professional mathematicians would agree. Sometimes you're surprised by what pops out when you turn the crank. (Witness: every non-trivial probability problem ever) Einstein's special theory of relativity was mainly an intuitive interpretation of Lorenz's discovery of Maxwell's equations' invariance under Lorenz transformation. The insight "c is constant in every frame of reference", would not have been possible without Lorenz mechanically working out what sort of transformation would leave Maxwell's equations invariant. Dirac predicted the existence of the positron solely based on a mechanical process of finding out what sort of equation satisfied the symmetries observed in nature. My point being that many great intellectual advances have been made by people who trusted the process more than themselves, and that's one of the cornerstones of mathematical thinking: trust the process more than yourself.
mwill 16y ago
Vi Harts youtube is pretty entertaining, Here's a video that shows pretty well what her videos are like: http://www.youtube.com/watch?v=Yhlv5Aeuo_k http://www.youtube.com/watch?v=Yhlv5Aeuo_k The whole topic vaguely reminds me of Khan Academy, both seem to recognize that the way math is taught in schools at the moment is fundamentally broken, and doesn't engage students the way it should.
rohanprabhu 16y ago
If taught properly, and with the proper illustrative exercises, calculus is recreational. Period.
- currywurst 16y ago
  Great point ! The "recreational" part can be applied to any subject imho.
- psykotic 16y ago
  Here's the thing. Most of calculus's power comes from its use as a mechanical calculational system. Hence the name. Archimedes's proof of the surface area and volume of a sphere is far more insightful and explanatory than the standard calculational proof using integration that any unthinking freshman can crank through with ease. As a rule, calculation proceeds by reducing a big problem into a sequence of small, easily dispatched problems, but human understanding usually works the opposite way. So, it's not that calculus cannot be taught in a way that highlights human understanding, but the greatest strength of calculus is that it produces answers whether you understand what you are doing or not.
amitagrawal 16y ago
The only reason Calculus seems so dry is that the ones teaching it are not very competent and don't understand the subject quite well. During my school I'd ask teachers all about limits and why we need it and their attitude was - Don't get too involved - here are the formulas, go learn them and solve the exercises. I later discovered that there were some amazing applications of limits (calculus) but wasn't taught the way they were meant to. This is the reason why I never found Calculus to be interesting.
crasshopper 16y ago
Every school should replace calculus with spreadsheets and a basic programming course. Or maybe with statistics. Each is more practical than calculus. (I don't mean to put down calculus; it's spiritually rewarding and, when paired with a few other subjects, yields crazy awesome higher math.)
- shou4577 16y ago
  While I would agree that a basic programming course might be helpful as a required course for all schools, I don't understand why it should replace Calculus? There is room for both, and they have different goals and advantages/disadvantages. You say yourself that calculus is awesome for several reasons - why do you think we should replace it? If it is only because a different course might be more practical, you have entered a slippery slope. College is not the same as trade school - we don't learn things because they are practical, but because there is an advantage simply to learning. I believe that all basic college courses should aim to promote critical thinking above all else, as the practicability of any given subject will vary widely between students.
  - crasshopper 16y ago
    why replace I'm just trying to be charitable to people who don't want music, drama, driver's ed, health class, ........ cut out. Why should recreational mathematics replace calculus, either? Ideally math would be taught well and written so that people could engage with it throughout life. (But how many adults do you see reading those yellow spiney Kluwer books at coffee shops?) slippery slope Thought Vi Hart was talking about high school. Maybe I was wrong. I agree that calculus is the right kind of brain-stretching for college.
    - shou4577 16y ago
      I agree completely. I don't think that recreational mathematics should replace calculus. I think that there are some definite changes that should be made to mathematics education, but that the majority of these should take place at a much younger age, even grade school. Early education is the perfect place for recreational mathematics. Fostering an interest at a young age (rather than force memorization of formulas, algorithms, and tables) can help a lot to keep people engaged throughout their lives. I think that by the time students reach the college level (which I think starts with Calculus, even if they reach this level in high school), they should be able to deal with some of the more boring parts of mathematics, sustained only by nuggets of usefulness and beauty. After all, not all mathematics is wondrous joy. I also agree that many of these problems could be fixed simply by teaching better. In calculus, for example, the beautiful thing is how closely it mirrors nature. Calculus states that if we know the position of an object at all times, we also know the velocity of the object at all times. This is a beautiful and fundamental result, one that I feel is not stressed enough during a first-year calculus course. I think that it is amazing not just that we know (in theory) the velocity, but that we can actually calculate it explicitly (in most circumstances) or approximate it numerically (in most circumstances). The parts of calculus on how to do that are somewhat less beautiful and paradigm-shifting, but are necessary, and adult students ought to be able to deal with that.
alain94040 16y ago
I know people here (the top 0.1% of the high school class) love to bash math education in particular. I'd like to ask how you would apply your views on teaching a topic such as History for instance. Sure, rather than learning the dates of middle-age battles, it would be more fun to interpret a play, in costume, or rebuild a battlefield with legos. But can you spend your entire school year this way, or could this "recreational" teaching need to be combined with more traditional teaching?
- crasshopper 16y ago
  Use primary sources. Have individuals or teams look at an original primary source, interpret it, then discuss with the class, who all interpret something from a related source. Students put their heads together, discuss/debate and post some interesting original research to the Net.
  - cdavid 16y ago
    primary sources is known to be one of the worst way to study any field, especially when you are not familiar with it, because you lack the context, and because they are rarely written for the profane. This is all the more true for maths: have you ever read a math books from the XIXth century ? This is almost impenetrable, if only because of the formulation
    - crasshopper 16y ago
      cdavid, I am talking about History. Pick up some letters from a soldier. The teacher can still give some context.
  - shou4577 16y ago
    In mathematics? Primary sources are way above the level of most college mathematics students, much less high school or younger students. Almost anything written in the last 50 years requires very advanced mathematical ideas, while anything written before that uses outdated notation which is very difficult to understand. Moreover, the primary sources for common ideas are often spread out over huge periods of refinements - going from a small idea to a general notion over the course of years, perhaps involving dozens of papers by multiple authors. One notable exception to this: Euler's papers. These might be readable by some students, yet when I see them presented in college courses, they are often presented by a secondary source for clarification. The other thing I don't agree with is your use of the word "interpret." I don't think that it is common to find a mathematician who believes that mathematics is open to interpretation. Lastly, original research in mathematics is hard. It's not the interpretation of previous ideas (although it is occasionally relating concepts previously thought to be unrelated - some consider this the most important type of mathematical result, but it is much harder to do than you might think), but the formulation of new ideas. Even to know if an idea is new requires a great deal of mathematical training. Perhaps I'm wrong about this, but I think that for young mathematics students your ideas are unfeasible. Remember that the original sources for a lot of basic mathematics are hundreds of years old. Even Euler's famous writings on geometry (the exception that I said might be workable) were largely a rewriting of previous ideas into a coherent whole - basically a textbook - and could not really be considered an original source. Edit: my apologies, I misunderstood. I did not realize that we were talking about teaching history, in which case I defer to someone of greater experience.
Swizec 16y ago
If you aren't familiar with Vi Hart, let me suggest some of the more awesome videos that seriously tickled my geeky innards. http://www.youtube.com/watch?v=DK5Z709J2eo http://www.youtube.com/watch?v=DK5Z709J2eo (infinite series through doodling) http://www.youtube.com/watch?v=heKK95DAKms http://www.youtube.com/watch?v=heKK95DAKms (graph theory awesomeness, through doodling) http://www.youtube.com/watch?v=e4MSN6IImpI http://www.youtube.com/watch?v=e4MSN6IImpI (some lovely properties of binary trees ... through doodling) http://www.youtube.com/watch?v=Gx5D09s5X6U http://www.youtube.com/watch?v=Gx5D09s5X6U (snakes) Especially that last one, it poses a question as to whether you can use a grammar to show that a snake lives (doesn't bite itself). Unfortunately she doesn't provide a solution to the problem, but me and a few mates went ahead and proved that it is impossible to prove whether the problem is solvable without simulating the snake.
RiderOfGiraffes 16y ago
When people start thinking about math education - as I have at considerable length - they always come up with the idea that since "real math" is about problem solving (which it is) and not about rote algorithms and manipulations (which it isn't) then what we should do is come up with problems to solve, not formulas to manipulate. We should encourage creative thinking, novel solutions, and ways of approaching problems. This is, actually, all true. Students should indulge in problem solving, analysis, investigation, early proof, ideas of proof, argument, explanation, and all the things that math is really about, and which make math useful. After all, who except for science and engineering students ever differentiate a polynomial? Knowning qualitatively and intuitively what rate-of-change really means is useful. Differentiating x.sin(x) isn't. But here's the problem. To within 20% or so life expectency in the UK is about 70, and there are about 70 million people, so roughly at every age point (up to about 50) there are about a million people. How do you assess and grade the mathematical standard (whatever that means) of about a million children? How do you persuade a government full of lawyers and classics graduates that what you're doing is better? How do you assess the teaching, and the school, if not by the grades they and their students achieve. It's all about assessment. So that's the first stumbling block. The second is the question of getting teachers who are up to the task of teaching in an open and interactive manner, rather than trying to teach rote algorithms and manipulations. But you don't need to worry about that - schools will never teach proper mathematics. They'll continue to teach arithmetic and manipluation.
- wisty 16y ago
  If you don't get the fundamentals down, you can't handle the applications. Wax on, wax off - boring but effective. But you also need to spend time putting those fundamentals to work. The real question is, why do you need assessment? To test that most kids are getting the fundamentals? Easy. Have a few core tests. To check that class time is also devoted to creative stuff, and not just drills? That's a management problem. You can't test whether or not kids have had (say) 50% of their time doing fun non-core work, but you can regulate that through other means.
  - RiderOfGiraffes 16y ago
mcdaid 16y ago
Well some of the videos look interesting but I don't agree with the title, what is recreational math? It would be great if Maths could be made more interesting and students make their own unique discoveries etc. However as a former maths teacher I can tell you that structure exists for a reason. In a class of maybe 30 students for one hour, the teacher can afford to give at best only 2 mins average time to each student. Some students may work very well independently, but others would take advantage of the freedom to mess about. Even if the students were focused how is a teacher with so little time per student supposed to pick up a random piece of work and then decide if the child is doing something of worth. The Maths curriculum needs to be rethought for the information age, but the solutions need to be practical.
- sid0 16y ago
  I think the Khan academy has a solution to this particular problem. Salman in his TED talk proposed that students view the videos in their own time for the basics, then engage in problem solving in class so that the teacher can devote much more attention to each student.
  - RiderOfGiraffes 16y ago
    Experience with this shows that the kids simply don't watch the videos. It sounds brilliant - let the kids watch videos together, they can eat pizza, drink cola, laugh, text, watch the video again to get the bits they missed, and then come to class to work the problems and get help. The experiment has been done - the vast majority of the kids don't watch the videos.
    - sid0 16y ago
      Is there a record of the experiment online somewhere?
      - RiderOfGiraffes 16y ago
        I'm pretty sure I read of it on a web site lunk to from here, but I don't have a record of it. I have seen it reported from at least two different people who tried it, but don't have a record of where I read it. Sorry. EDIT: I've found one reference to a teacher who's trying it but has some kids who don't watch the videos: http://www.msteacher2.org/forum/topics/flipping-the-math-classroom?xg_source=activity http://www.msteacher2.org/forum/topics/flipping-the-math-cla... Here's another in the comment stream for the Khan TED talk that suggests the idea: http://www.joannejacobs.com/2011/03/khan-use-video-to-flip-education/comment-page-1/#comment-159231 http://www.joannejacobs.com/2011/03/khan-use-video-to-flip-e... There are more, but they require filtering/finding: http://www.google.co.uk/search?q=math+homework+flip+video+%22don%27t+watch%22 http://www.google.co.uk/search?q=math+homework+flip+video+%2...
Tycho 16y ago
Incidentally there is a textbook on calculus that Donald Knuth said he read in his first year at university, and after doing all the problems in it he moved from beginner to top of the class (and thereafter, renowned computing scientist). I think the author was named Thompson, but I haven't seen the book myself. Maybe some people here have?
- akuchling 16y ago
  Maybe it was "Calculus Made Easy", by Silvanus P. Thompson? That's an old book that explains the basic idea of integration and differentiation, but doesn't try to be rigorous at all. http://www.scribd.com/doc/8533492/Calculus-Made-Easy-by-Silvanus-P-Thompson http://www.scribd.com/doc/8533492/Calculus-Made-Easy-by-Silv...
  - hsmyers 16y ago
    Non scribe D version: http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf
ANH 16y ago
Freeman Dyson wrote this of Richard Feynman: "The reason Dick's physics was so hard for ordinary people to grasp was that he did not use equations. The usual way theoretical physics was done since the time of Newton was to begin by writing down some equations and then to work hard calculating solutions of the equations ... Dick just wrote down the solutions out of his head without ever writing down the equations. He had a physical picture of the way things happen, and the picture gave him the solutions directly with a minimum of calculation." edit: That's from Dyson's autobiography 'Disturbing the Universe'.
- HSO 16y ago
  Which is interesting and awesome but so not relevant for general public math education... ;-)
- zeynel1 16y ago
  "The usual way theoretical physics was done since the time of Newton was to begin by writing down some equations and then to work hard calculating solutions of the equations" I cannot believe that Freeman Dyson would claim such false notion that Newton worked with "equations." I checked the quote and http://books.google.com/books?id=RHzoMeU2bxsC&printsec=frontcover&dq=disturbing+the+universe&hl=en&ei=xuyUTcX_GsGa0QHSsoGKDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA#v=snippet&q=newton&f=false http://books.google.com/books?id=RHzoMeU2bxsC&printsec=f... it is true, the quote is in the book. I would think that Dyson would know that Newton did not write one single equation ever. He worked with proportions.
HSO 16y ago
I'm really not convinced by this. Sure, math education is far from ideal. But the solution cannot be to make it more "touchy-feely" and make everybody feel good. Courant, in his book "What is mathematics?", says the field historically seems to go through alternating phases, from wild, creative, adventurous phases in which new things are discovered -- often inspired by "real-world" problems in physics or engineering -- to more sober, rigorous phases in which another generation takes over and basically cleans up the findings of their predecessors (making proper definitions, constructing and refining the proofs etc.). But this is far from what math _education_ can be about. The biggest obstacle to math education is the meme that mathematics is "only for intelligent people". Mathematics at the basic, cleaned up level has almost nothing to do with "intelligence" and a lot to do with memory. I know this first-hand because I used to be really bad at maths and then one day decided to learn it anyway (and did). Most often, when someone doesn't get some result or can't figure out the derivation, it's because he doesn't remember or has never learned the relevant facts. [EDIT: Also, if you think you need to be "intelligent" (however defined) to do math, you will give up sooner when you don't understand sth because you think "uh, I'm too dumb" instead of asking "what am I missing".] Mathematics is a kind of language, and as in any language, most people will actually need to sit down and memorize vocabulary. It's mundane, it's not fancy-creative, it's a pain in your behind, but once you succeed in automating certain things, the process takes on a momentum of its own. This "dragon mom" stuff I occasionally read about in the NYT is silly IMHO but it's just as silly for a 22 year-old, who by her own admission doesn't even know the standard curriculum of a plain vanilla mathematics education, to want to be "the ambassador of mathematics" (http://www.nytimes.com/2011/01/18/science/18prof.html http://www.nytimes.com/2011/01/18/science/18prof.html). First you do the hard work, then you can fly.
- sp332 16y ago
  Math education should at the very least include a description of what math is and why it's interesting. Current math curricula don't let students know that math can be engaging, and so they're not engaged. The ambassador thing is to tell "normal" people what math is about. That's fundamental, not "touchy-feely".
  - 16y ago
mtalantikite 16y ago
When I was in school majoring in math we always thought back to how poorly motivated the subject was growing up. So we decided to explain some pretty basic group theory (permutations, etc) to our friend's dad who taught elementary school. When he brought in the topic with some basic shapes to illustrate the concepts to his kids (4th grade, I think), they all went crazy for it. We then gave him other concepts from a geometry class, which they loved as well (http://www.gang.umass.edu/~kusner/class/462hw http://www.gang.umass.edu/~kusner/class/462hw). I think catching them at an early age -- sparking their natural curiosity and giving them confidence in the subject -- is most important, before they're disheartened by all the years of computation and applying algorithms.
ihumanable 16y ago
I had no interest in math until I had Calculus in High School. I then went on to pursue dual degrees in Computer Science (because I <3 Programming) and Pure Mathematics (because Calculus made me <3 Math). Maybe it was just an exceptionally good teacher, but the more I learned about Calculus the more I feel it was the subject itself. There was something seductive in the idea that by changing the way you think about something, like the area under a curve, you could calculate the previously impossible. The core of Calculus (at least the part you would learn in High School), derivatives and integrals, are easily enough explained. I'm glad I had Calculus or I wouldn't have gone on to pursue mathematics so doggedly, I wouldn't have gotten to learn the strange and beautiful world of numbers, I may have missed out on one of my great loves. That said, I'm sure in the hands of a less than stellar teacher, Calculus is a painful thing.
sp332 16y ago
The head of the math department at my college actually taught one of these classes. He would assign two students each class period to take notes. He said that way, the rest of us could pay attention :) He would stand at the front and ramble about whatever came to mind, and scribble on an overhead transparency. He would start with a topic and take it wherever we asked: multiple dimensions, knot-tying, origami, computational geometry, ray-tracing. The course was called "3D Math" or something, so everything was relevant :) There were no tests, and we got to choose our own final projects. The notes and overheads were duplicated as study guides for the final, and the questions were taken right out of them. It was the most informative math class, if only because we were all doing our own thing and learning from each other. I did a 3D extension of the Lyapunov fractal images that Mario Markus published in Scientific American in 1992. Markus's 2D: http://charles.vassallo.pagesperso-orange.fr/en/lyap_art/lyapdoc.html http://charles.vassallo.pagesperso-orange.fr/en/lyap_art/lya... and my 3D http://www.youtube.com/watch?v=-omN5ZM3Jho http://www.youtube.com/watch?v=-omN5ZM3Jho with a moving cross-section http://www.youtube.com/watch?v=ePRJfF3pwqg http://www.youtube.com/watch?v=ePRJfF3pwqg
duck 16y ago
I have to ask... does the name Vi come from where I think it does? If so... that is awesome, and then I would have to ask is there a improved version of her named Vim?
upgrayedd 16y ago
Everyone here on HN always seems to say the same thing: the way math is being taught in most schools sucks, we need to change it. Math is glorious, it's the kicks, etc. I'm curious why everyone is so enamored with mathematics here though; I've tried to understand what is so captivating about the subject but have failed to do so for years. I've never gotten past trigonometry and algebra myself, and this idea of Calculus being more than the tedious integration work that textbooks I tried to self teach from said it was is intriguing. Enlighten me, how is mathematics not just a bunch of rules and methods to solve problems within constraints, and that some professional practitioners are forced to prove... How many of you programmers and software engineers here even interact with the Calculus you learned in high school/college on a day-to-day basis in your apps/software projects?
forkandwait 16y ago
Just one question -- does the US have a shortage of engineers because math education isn't touchy-feely enough now?
anactofgod 16y ago
"You don't actually need to know a lot of the groundwork and the basics to appreciate the exciting bits of mathematics. The more technical things are still awesome as tools and are necessary in some places. But luckily I am usually with other people who do know that stuff." How about this not being an either/or? Anything that exposes more people to the beauty and power of the maths, and convinces some of them to more of the "technical things" is goodness, IMHO.

It's so true that the way that maths is taught at schools needs a complete revamp. At secondary school level it should be taught as a creative subject from the start: choose some rules and see what happens.

xyzzyz 16y ago

I'm a Math and CS student and still wonder how was it even possible that I did not start to hate math in high school. Things done there are dull and boring, exercises are repetitive and do not teach anything substantial other than applying another algorithm shown by a teacher. I found out that almost everyone did not really understand not only the problems that were told to solve, but also the solutions they gave -- and it was hard to blame them, for it was mainly teacher's and education system's fault not to give them thorough understanding of what is math really about. My idea? Dump all that algorithmic crap and teach real problem solving. Do not punish for bad performance, because it is really hard to actually come up with a solution to a problem, and this skill is not something that can be learned by doing the same thing over and over again. Problem solving also require less concepts to be introduced and internalized, so more time can be spent on actual thinking and discussing, and less on preparations. This kind of math course may seem not to prepare to good to "university level" math, but in fact it is quite opposite -- universities have to start almost from scratch anyway, and it is easier to teach new concepts a person who has a good understanding what math really is, than a person who only know tens of algorithms, used in high school to grind all kind of "problems" given there.

btam 16y ago
The thing is, students already think of math as difficult. If we switched from algorithms to actual creative problem solving, it would become a lot more intimidating. I'm pretty confident of this. I'm in high school, and the other students have noted how much more they would 'hate' math if it were like the creative problem solving math competition that we have here in the US each year. http://en.wikipedia.org/wiki/American_Mathematics_Competitions http://en.wikipedia.org/wiki/American_Mathematics_Competitio... What many -- perhaps even most -- kids really want is a course where they do only simple mechanical work, and get a boost to their GPA. At least, that's what it seems to me to be. Also, we have to consider that unfortunately, not all teachers care. Now that teaching to the test is rampant(at least in the US), I think many classes would fall apart if the added structure and 'accountability' disappeared, like it would if we made that switch. I don't disagree with you. The world would be a better place if we could get rid of this; if we didn't have all of this baggage to deal with. EDIT: RiderOfGiraffes seems to have said the same thing in a different way in this thread; check it out: http://news.ycombinator.com/item?id=2390960 http://news.ycombinator.com/item?id=2390960.
- RiderOfGiraffes 16y ago
  But that's math competition stuff, which is seriously more daunting, and practising mathematicians will often claim is even less like "real, proper math." It's occasionally suggested that some real, long-standing unsolved problems should be set as IMO questions, because there's a non-zero chance they'd get solved. > What many -- perhaps even most -- kids really want is a > course where they do only simple mechanical work, and > get a boost to their GPA. "Most people would sooner die than think. And most of them do." -- G.B.Shaw. > ... not all teachers care. And not all teachers are capable. And largely all of your points are exactly right, except for your misconception about what kind of math we want kids to be doing. Going back to that point, the IMO is like playing at Carnegie Hall. What we want is like messing about with instruments to see what they do, and getting hooked on the curious things that are possible. I'm not offering solutions here, I'm just helping define the problem, and explore directions.
  - btam 16y ago
    You're pretty much right on the math competitions front, but the first few problems on the AMC are pretty simple IMO(pardon the pun). They are barely above the level of mechanical problem solving. It does get harder, though, and I agree that that is more daunting. You've reminded me, however -- I remembered thinking that the Phillips Exeter Academy model for math education (the "Harkness" method) is amazing. http://en.wikipedia.org/wiki/Phillips_Exeter_Academy#Harkness_teaching_method http://en.wikipedia.org/wiki/Phillips_Exeter_Academy#Harknes... > Exeter does not teach math with traditional textbooks. Instead, > math teachers assign problems from workbooks that have been > written collectively by the Academy's math department. From these > custom workbooks, students are assigned word problems as > homework. In class, students then present their solutions at the > blackboard. This means that in math class at Exeter, students are not > given theorems, model problems, or principles beforehand. Instead, > theorems and principles emerge more organically, as students work > through the word problems."* This is what we want. Discovering math makes it fun for everyone. Honesty and actual learning with that method(my whole would be not be too difficult, because everyone would inevitably have their own personal spin on the origins of the theorems et cetera. Plus, it'd be embarrassing to go to the blackboard and say that you haven't done the work. ---- By the way, expect an email regarding what you said about the IMO.
crasshopper 16y ago
Lockhart's Lament: www.maa.org/devlin/LockhartsLament.pdf A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory.... Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music ... are considered very advanced topics and are generally put off until college, and more often graduate school.
samlittlewood 16y ago
The incentives should be around changes in performance, +ve or -ve, especially if it was as a result of the kid's own actions.
icegreentea 16y ago
From talking to my professors, there really was once a time when university math (calc 101 or w/e) was not starting from scratch (from the perspective of engineering anyways). Spend maybe a month over derivatives and limits, and then off to integrals and everything else you'll need so that mechanics actually makes some sense.
- deleted 16y ago
  [deleted]
neutronicus 16y ago
I'm probably one of the few, but I didn't have any problem with the paradigm of high school math education, just the pace. As far as my later career, it's been my experience that rote manipulation without conceptual understanding can solve your problem, and that the converse isn't true. Sometimes, if you do the math right, it will contradict your intuition. I'm hesitant to endorse any paradigm that aims to make math more intuitive, since at the end of the day that's a lie. One of the take-home messages of mathematics education should be that these mechanical processes are better than your own "problem-solving skills".
- eru 16y ago
  In some sense, yes. Mathematics is the science of how to solve problems without thinking. Progress in mathematics means elimination of thought needed. But actually inventing that time saving machinery is different from applying it.
  - ajscherer 16y ago
    Mathematics is most definitley not that. The fact that your definition probably sounds reasonable to most people who made it all the way through calculus (and even excelled in those courses) probably supports what Vi is saying. On the other hand, maybe it's okay that people don't really understand what math is (as mathematicians understand it). I don't see any evidence that society is falling apart due to lack of appreciation for pure math.
    - neutronicus 16y ago
      I think "mathematics is a mechanical process for solving problems without an intuitive grasp of their nature" is a very good definition, with which many professional mathematicians would agree. Sometimes you're surprised by what pops out when you turn the crank. (Witness: every non-trivial probability problem ever) Einstein's special theory of relativity was mainly an intuitive interpretation of Lorenz's discovery of Maxwell's equations' invariance under Lorenz transformation. The insight "c is constant in every frame of reference", would not have been possible without Lorenz mechanically working out what sort of transformation would leave Maxwell's equations invariant. Dirac predicted the existence of the positron solely based on a mechanical process of finding out what sort of equation satisfied the symmetries observed in nature. My point being that many great intellectual advances have been made by people who trusted the process more than themselves, and that's one of the cornerstones of mathematical thinking: trust the process more than yourself.

mwill 16y ago

Vi Harts youtube is pretty entertaining, Here's a video that shows pretty well what her videos are like: http://www.youtube.com/watch?v=Yhlv5Aeuo_k http://www.youtube.com/watch?v=Yhlv5Aeuo_k The whole topic vaguely reminds me of Khan Academy, both seem to recognize that the way math is taught in schools at the moment is fundamentally broken, and doesn't engage students the way it should.

rohanprabhu 16y ago

If taught properly, and with the proper illustrative exercises, calculus is recreational. Period.

currywurst 16y ago
Great point ! The "recreational" part can be applied to any subject imho.
psykotic 16y ago
Here's the thing. Most of calculus's power comes from its use as a mechanical calculational system. Hence the name. Archimedes's proof of the surface area and volume of a sphere is far more insightful and explanatory than the standard calculational proof using integration that any unthinking freshman can crank through with ease. As a rule, calculation proceeds by reducing a big problem into a sequence of small, easily dispatched problems, but human understanding usually works the opposite way. So, it's not that calculus cannot be taught in a way that highlights human understanding, but the greatest strength of calculus is that it produces answers whether you understand what you are doing or not.

amitagrawal 16y ago

The only reason Calculus seems so dry is that the ones teaching it are not very competent and don't understand the subject quite well. During my school I'd ask teachers all about limits and why we need it and their attitude was - Don't get too involved - here are the formulas, go learn them and solve the exercises. I later discovered that there were some amazing applications of limits (calculus) but wasn't taught the way they were meant to. This is the reason why I never found Calculus to be interesting.

crasshopper 16y ago

Every school should replace calculus with spreadsheets and a basic programming course. Or maybe with statistics. Each is more practical than calculus. (I don't mean to put down calculus; it's spiritually rewarding and, when paired with a few other subjects, yields crazy awesome higher math.)

shou4577 16y ago
While I would agree that a basic programming course might be helpful as a required course for all schools, I don't understand why it should replace Calculus? There is room for both, and they have different goals and advantages/disadvantages. You say yourself that calculus is awesome for several reasons - why do you think we should replace it? If it is only because a different course might be more practical, you have entered a slippery slope. College is not the same as trade school - we don't learn things because they are practical, but because there is an advantage simply to learning. I believe that all basic college courses should aim to promote critical thinking above all else, as the practicability of any given subject will vary widely between students.
- crasshopper 16y ago
  why replace I'm just trying to be charitable to people who don't want music, drama, driver's ed, health class, ........ cut out. Why should recreational mathematics replace calculus, either? Ideally math would be taught well and written so that people could engage with it throughout life. (But how many adults do you see reading those yellow spiney Kluwer books at coffee shops?) slippery slope Thought Vi Hart was talking about high school. Maybe I was wrong. I agree that calculus is the right kind of brain-stretching for college.
  - shou4577 16y ago
    I agree completely. I don't think that recreational mathematics should replace calculus. I think that there are some definite changes that should be made to mathematics education, but that the majority of these should take place at a much younger age, even grade school. Early education is the perfect place for recreational mathematics. Fostering an interest at a young age (rather than force memorization of formulas, algorithms, and tables) can help a lot to keep people engaged throughout their lives. I think that by the time students reach the college level (which I think starts with Calculus, even if they reach this level in high school), they should be able to deal with some of the more boring parts of mathematics, sustained only by nuggets of usefulness and beauty. After all, not all mathematics is wondrous joy. I also agree that many of these problems could be fixed simply by teaching better. In calculus, for example, the beautiful thing is how closely it mirrors nature. Calculus states that if we know the position of an object at all times, we also know the velocity of the object at all times. This is a beautiful and fundamental result, one that I feel is not stressed enough during a first-year calculus course. I think that it is amazing not just that we know (in theory) the velocity, but that we can actually calculate it explicitly (in most circumstances) or approximate it numerically (in most circumstances). The parts of calculus on how to do that are somewhat less beautiful and paradigm-shifting, but are necessary, and adult students ought to be able to deal with that.

alain94040 16y ago

I know people here (the top 0.1% of the high school class) love to bash math education in particular. I'd like to ask how you would apply your views on teaching a topic such as History for instance. Sure, rather than learning the dates of middle-age battles, it would be more fun to interpret a play, in costume, or rebuild a battlefield with legos. But can you spend your entire school year this way, or could this "recreational" teaching need to be combined with more traditional teaching?

crasshopper 16y ago
Use primary sources. Have individuals or teams look at an original primary source, interpret it, then discuss with the class, who all interpret something from a related source. Students put their heads together, discuss/debate and post some interesting original research to the Net.
- cdavid 16y ago
  primary sources is known to be one of the worst way to study any field, especially when you are not familiar with it, because you lack the context, and because they are rarely written for the profane. This is all the more true for maths: have you ever read a math books from the XIXth century ? This is almost impenetrable, if only because of the formulation
  - crasshopper 16y ago
    cdavid, I am talking about History. Pick up some letters from a soldier. The teacher can still give some context.
- shou4577 16y ago
  In mathematics? Primary sources are way above the level of most college mathematics students, much less high school or younger students. Almost anything written in the last 50 years requires very advanced mathematical ideas, while anything written before that uses outdated notation which is very difficult to understand. Moreover, the primary sources for common ideas are often spread out over huge periods of refinements - going from a small idea to a general notion over the course of years, perhaps involving dozens of papers by multiple authors. One notable exception to this: Euler's papers. These might be readable by some students, yet when I see them presented in college courses, they are often presented by a secondary source for clarification. The other thing I don't agree with is your use of the word "interpret." I don't think that it is common to find a mathematician who believes that mathematics is open to interpretation. Lastly, original research in mathematics is hard. It's not the interpretation of previous ideas (although it is occasionally relating concepts previously thought to be unrelated - some consider this the most important type of mathematical result, but it is much harder to do than you might think), but the formulation of new ideas. Even to know if an idea is new requires a great deal of mathematical training. Perhaps I'm wrong about this, but I think that for young mathematics students your ideas are unfeasible. Remember that the original sources for a lot of basic mathematics are hundreds of years old. Even Euler's famous writings on geometry (the exception that I said might be workable) were largely a rewriting of previous ideas into a coherent whole - basically a textbook - and could not really be considered an original source. Edit: my apologies, I misunderstood. I did not realize that we were talking about teaching history, in which case I defer to someone of greater experience.

Swizec 16y ago

If you aren't familiar with Vi Hart, let me suggest some of the more awesome videos that seriously tickled my geeky innards. http://www.youtube.com/watch?v=DK5Z709J2eo http://www.youtube.com/watch?v=DK5Z709J2eo (infinite series through doodling) http://www.youtube.com/watch?v=heKK95DAKms http://www.youtube.com/watch?v=heKK95DAKms (graph theory awesomeness, through doodling) http://www.youtube.com/watch?v=e4MSN6IImpI http://www.youtube.com/watch?v=e4MSN6IImpI (some lovely properties of binary trees ... through doodling) http://www.youtube.com/watch?v=Gx5D09s5X6U http://www.youtube.com/watch?v=Gx5D09s5X6U (snakes) Especially that last one, it poses a question as to whether you can use a grammar to show that a snake lives (doesn't bite itself). Unfortunately she doesn't provide a solution to the problem, but me and a few mates went ahead and proved that it is impossible to prove whether the problem is solvable without simulating the snake.

RiderOfGiraffes 16y ago

When people start thinking about math education - as I have at considerable length - they always come up with the idea that since "real math" is about problem solving (which it is) and not about rote algorithms and manipulations (which it isn't) then what we should do is come up with problems to solve, not formulas to manipulate. We should encourage creative thinking, novel solutions, and ways of approaching problems. This is, actually, all true. Students should indulge in problem solving, analysis, investigation, early proof, ideas of proof, argument, explanation, and all the things that math is really about, and which make math useful. After all, who except for science and engineering students ever differentiate a polynomial? Knowning qualitatively and intuitively what rate-of-change really means is useful. Differentiating x.sin(x) isn't. But here's the problem. To within 20% or so life expectency in the UK is about 70, and there are about 70 million people, so roughly at every age point (up to about 50) there are about a million people. How do you assess and grade the mathematical standard (whatever that means) of about a million children? How do you persuade a government full of lawyers and classics graduates that what you're doing is better? How do you assess the teaching, and the school, if not by the grades they and their students achieve. It's all about assessment. So that's the first stumbling block. The second is the question of getting teachers who are up to the task of teaching in an open and interactive manner, rather than trying to teach rote algorithms and manipulations. But you don't need to worry about that - schools will never teach proper mathematics. They'll continue to teach arithmetic and manipluation.

wisty 16y ago
If you don't get the fundamentals down, you can't handle the applications. Wax on, wax off - boring but effective. But you also need to spend time putting those fundamentals to work. The real question is, why do you need assessment? To test that most kids are getting the fundamentals? Easy. Have a few core tests. To check that class time is also devoted to creative stuff, and not just drills? That's a management problem. You can't test whether or not kids have had (say) 50% of their time doing fun non-core work, but you can regulate that through other means.
- RiderOfGiraffes 16y ago

mcdaid 16y ago

Well some of the videos look interesting but I don't agree with the title, what is recreational math? It would be great if Maths could be made more interesting and students make their own unique discoveries etc. However as a former maths teacher I can tell you that structure exists for a reason. In a class of maybe 30 students for one hour, the teacher can afford to give at best only 2 mins average time to each student. Some students may work very well independently, but others would take advantage of the freedom to mess about. Even if the students were focused how is a teacher with so little time per student supposed to pick up a random piece of work and then decide if the child is doing something of worth. The Maths curriculum needs to be rethought for the information age, but the solutions need to be practical.

sid0 16y ago
I think the Khan academy has a solution to this particular problem. Salman in his TED talk proposed that students view the videos in their own time for the basics, then engage in problem solving in class so that the teacher can devote much more attention to each student.
- RiderOfGiraffes 16y ago
  Experience with this shows that the kids simply don't watch the videos. It sounds brilliant - let the kids watch videos together, they can eat pizza, drink cola, laugh, text, watch the video again to get the bits they missed, and then come to class to work the problems and get help. The experiment has been done - the vast majority of the kids don't watch the videos.
  - sid0 16y ago
    Is there a record of the experiment online somewhere?
    - RiderOfGiraffes 16y ago
      I'm pretty sure I read of it on a web site lunk to from here, but I don't have a record of it. I have seen it reported from at least two different people who tried it, but don't have a record of where I read it. Sorry. EDIT: I've found one reference to a teacher who's trying it but has some kids who don't watch the videos: http://www.msteacher2.org/forum/topics/flipping-the-math-classroom?xg_source=activity http://www.msteacher2.org/forum/topics/flipping-the-math-cla... Here's another in the comment stream for the Khan TED talk that suggests the idea: http://www.joannejacobs.com/2011/03/khan-use-video-to-flip-education/comment-page-1/#comment-159231 http://www.joannejacobs.com/2011/03/khan-use-video-to-flip-e... There are more, but they require filtering/finding: http://www.google.co.uk/search?q=math+homework+flip+video+%22don%27t+watch%22 http://www.google.co.uk/search?q=math+homework+flip+video+%2...

Tycho 16y ago

Incidentally there is a textbook on calculus that Donald Knuth said he read in his first year at university, and after doing all the problems in it he moved from beginner to top of the class (and thereafter, renowned computing scientist). I think the author was named Thompson, but I haven't seen the book myself. Maybe some people here have?

akuchling 16y ago
Maybe it was "Calculus Made Easy", by Silvanus P. Thompson? That's an old book that explains the basic idea of integration and differentiation, but doesn't try to be rigorous at all. http://www.scribd.com/doc/8533492/Calculus-Made-Easy-by-Silvanus-P-Thompson http://www.scribd.com/doc/8533492/Calculus-Made-Easy-by-Silv...
- hsmyers 16y ago
  Non scribe D version: http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf

ANH 16y ago

Freeman Dyson wrote this of Richard Feynman: "The reason Dick's physics was so hard for ordinary people to grasp was that he did not use equations. The usual way theoretical physics was done since the time of Newton was to begin by writing down some equations and then to work hard calculating solutions of the equations ... Dick just wrote down the solutions out of his head without ever writing down the equations. He had a physical picture of the way things happen, and the picture gave him the solutions directly with a minimum of calculation." edit: That's from Dyson's autobiography 'Disturbing the Universe'.

HSO 16y ago
Which is interesting and awesome but so not relevant for general public math education... ;-)
zeynel1 16y ago
"The usual way theoretical physics was done since the time of Newton was to begin by writing down some equations and then to work hard calculating solutions of the equations" I cannot believe that Freeman Dyson would claim such false notion that Newton worked with "equations." I checked the quote and http://books.google.com/books?id=RHzoMeU2bxsC&printsec=frontcover&dq=disturbing+the+universe&hl=en&ei=xuyUTcX_GsGa0QHSsoGKDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA#v=snippet&q=newton&f=false http://books.google.com/books?id=RHzoMeU2bxsC&printsec=f... it is true, the quote is in the book. I would think that Dyson would know that Newton did not write one single equation ever. He worked with proportions.

HSO 16y ago

I'm really not convinced by this. Sure, math education is far from ideal. But the solution cannot be to make it more "touchy-feely" and make everybody feel good. Courant, in his book "What is mathematics?", says the field historically seems to go through alternating phases, from wild, creative, adventurous phases in which new things are discovered -- often inspired by "real-world" problems in physics or engineering -- to more sober, rigorous phases in which another generation takes over and basically cleans up the findings of their predecessors (making proper definitions, constructing and refining the proofs etc.). But this is far from what math _education_ can be about. The biggest obstacle to math education is the meme that mathematics is "only for intelligent people". Mathematics at the basic, cleaned up level has almost nothing to do with "intelligence" and a lot to do with memory. I know this first-hand because I used to be really bad at maths and then one day decided to learn it anyway (and did). Most often, when someone doesn't get some result or can't figure out the derivation, it's because he doesn't remember or has never learned the relevant facts. [EDIT: Also, if you think you need to be "intelligent" (however defined) to do math, you will give up sooner when you don't understand sth because you think "uh, I'm too dumb" instead of asking "what am I missing".] Mathematics is a kind of language, and as in any language, most people will actually need to sit down and memorize vocabulary. It's mundane, it's not fancy-creative, it's a pain in your behind, but once you succeed in automating certain things, the process takes on a momentum of its own. This "dragon mom" stuff I occasionally read about in the NYT is silly IMHO but it's just as silly for a 22 year-old, who by her own admission doesn't even know the standard curriculum of a plain vanilla mathematics education, to want to be "the ambassador of mathematics" (http://www.nytimes.com/2011/01/18/science/18prof.html http://www.nytimes.com/2011/01/18/science/18prof.html). First you do the hard work, then you can fly.

sp332 16y ago
Math education should at the very least include a description of what math is and why it's interesting. Current math curricula don't let students know that math can be engaging, and so they're not engaged. The ambassador thing is to tell "normal" people what math is about. That's fundamental, not "touchy-feely".
- 16y ago

mtalantikite 16y ago

When I was in school majoring in math we always thought back to how poorly motivated the subject was growing up. So we decided to explain some pretty basic group theory (permutations, etc) to our friend's dad who taught elementary school. When he brought in the topic with some basic shapes to illustrate the concepts to his kids (4th grade, I think), they all went crazy for it. We then gave him other concepts from a geometry class, which they loved as well (http://www.gang.umass.edu/~kusner/class/462hw http://www.gang.umass.edu/~kusner/class/462hw). I think catching them at an early age -- sparking their natural curiosity and giving them confidence in the subject -- is most important, before they're disheartened by all the years of computation and applying algorithms.

ihumanable 16y ago

I had no interest in math until I had Calculus in High School. I then went on to pursue dual degrees in Computer Science (because I <3 Programming) and Pure Mathematics (because Calculus made me <3 Math). Maybe it was just an exceptionally good teacher, but the more I learned about Calculus the more I feel it was the subject itself. There was something seductive in the idea that by changing the way you think about something, like the area under a curve, you could calculate the previously impossible. The core of Calculus (at least the part you would learn in High School), derivatives and integrals, are easily enough explained. I'm glad I had Calculus or I wouldn't have gone on to pursue mathematics so doggedly, I wouldn't have gotten to learn the strange and beautiful world of numbers, I may have missed out on one of my great loves. That said, I'm sure in the hands of a less than stellar teacher, Calculus is a painful thing.

sp332 16y ago

The head of the math department at my college actually taught one of these classes. He would assign two students each class period to take notes. He said that way, the rest of us could pay attention :) He would stand at the front and ramble about whatever came to mind, and scribble on an overhead transparency. He would start with a topic and take it wherever we asked: multiple dimensions, knot-tying, origami, computational geometry, ray-tracing. The course was called "3D Math" or something, so everything was relevant :) There were no tests, and we got to choose our own final projects. The notes and overheads were duplicated as study guides for the final, and the questions were taken right out of them. It was the most informative math class, if only because we were all doing our own thing and learning from each other. I did a 3D extension of the Lyapunov fractal images that Mario Markus published in Scientific American in 1992. Markus's 2D: http://charles.vassallo.pagesperso-orange.fr/en/lyap_art/lyapdoc.html http://charles.vassallo.pagesperso-orange.fr/en/lyap_art/lya... and my 3D http://www.youtube.com/watch?v=-omN5ZM3Jho http://www.youtube.com/watch?v=-omN5ZM3Jho with a moving cross-section http://www.youtube.com/watch?v=ePRJfF3pwqg http://www.youtube.com/watch?v=ePRJfF3pwqg

duck 16y ago

I have to ask... does the name Vi come from where I think it does? If so... that is awesome, and then I would have to ask is there a improved version of her named Vim?

upgrayedd 16y ago

Everyone here on HN always seems to say the same thing: the way math is being taught in most schools sucks, we need to change it. Math is glorious, it's the kicks, etc. I'm curious why everyone is so enamored with mathematics here though; I've tried to understand what is so captivating about the subject but have failed to do so for years. I've never gotten past trigonometry and algebra myself, and this idea of Calculus being more than the tedious integration work that textbooks I tried to self teach from said it was is intriguing. Enlighten me, how is mathematics not just a bunch of rules and methods to solve problems within constraints, and that some professional practitioners are forced to prove... How many of you programmers and software engineers here even interact with the Calculus you learned in high school/college on a day-to-day basis in your apps/software projects?

forkandwait 16y ago

Just one question -- does the US have a shortage of engineers because math education isn't touchy-feely enough now?

anactofgod 16y ago

"You don't actually need to know a lot of the groundwork and the basics to appreciate the exciting bits of mathematics. The more technical things are still awesome as tools and are necessary in some places. But luckily I am usually with other people who do know that stuff." How about this not being an either/or? Anything that exposes more people to the beauty and power of the maths, and convinces some of them to more of the "technical things" is goodness, IMHO.